A prophet inequality for L2-martingales

Let k=(kn)n≥1 be the sequence given by the conditions k1=0 and kn+1=(1+kn2)/2, n≥1. We prove that for any L2-martingale X=(X1,X2,…,Xn) we have Emax1≤k≤nXk≤supτEXτ+knmax1≤k≤nV arXk, where the supremum on the right is taken over all stopping times τ of X which are bounded by n. Furthermore, it is shown that for each n, the constant kn is the best possible.